On the structure of the power graph and the enhanced power graph of a group

TL;DR

This paper explores the structure of power and enhanced power graphs of groups, characterizing their properties, especially for nilpotent and bounded exponent groups, and compares them with commuting graphs.

Contribution

It introduces the enhanced power graph, characterizes when power graphs are perfect, and compares these graphs with commuting graphs for finite groups.

Findings

Power graphs of bounded exponent groups are perfect.

Finite groups where power and enhanced power graphs are equal are characterized.

The clique number of power graphs is at most countably infinite.

Abstract

Let $G$ be a group. The \emph{power graph} of $G$ is a graph with the vertex set $G$, having an edge between two elements whenever one is a power of the other. We characterize nilpotent groups whose power graphs have finite independence number. For a bounded exponent group, we prove its power graph is a perfect graph and we determine its clique/chromatic number. Furthermore, it is proved that for every group $G$, the clique number of the power graph of $G$ is at most countably infinite. We also measure how close the power graph is to the \emph{commuting graph} by introducing a new graph which lies in between. We call this new graph as the \emph{enhanced power graph}. For an arbitrary pair of these three graphs we characterize finite groups for which this pair of graphs are equal.